How to sketch a graph of a function. Draw a sketch of the graph of the function, knowing that. Protection of personal information

“Derivative problems” - ?f(x) = f(x) - f(x0). x0 x0+?x. How do you imagine instantaneous speed? Instantaneous velocity problem. y. How do you imagine instantaneous speed? ?X=x-x0. What has been said is written down in the form. First, we defined the “territory” of our research. A l g o r i t m. The speed v gradually increases.

“Study of the derivative function” - The cannon fires at an angle to the horizon. Option 1 A B D Option 2 G B B. Municipal Educational Institution Meshkovskaya Secondary School Mathematics teacher Kovaleva T.V. The function is defined on the segment [-4;4] . How are derivative and function related? Answers: APPLYING THE DERIVATIVE TO THE STUDY OF THE FUNCTION: increasing and decreasing functions. TASK Remember the story about Baron Munchausen?

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“Application of the derivative to the study of functions” - 6. -1. 8. Identify the critical points of the function using the graph of the derivative of the function. 1. =. July 1, 1646 - November 14, 1716, Warm-up. A sign of increasing and decreasing function. Determine the sign of the derivative of the function on intervals.

“Lesson on the derivative of a complex function” - The derivative of a complex function. Calculate the speed of the point: a) at time t; b) at the moment t=2 s. Find the derivatives of the functions: , If. Brooke Taylor. Find the differential of the function: At what values ​​of x does the equality hold. The point moves rectilinearly according to the law s(t) = s(t) = (s is the path in meters, t is time in seconds).

“Definition of derivative” - 1. Proof: f(x+ ?x). Let u(x), v(x) and w(x) be differentiable functions in some interval (a; b), C is a constant. f(x). Equation of a straight line with an angular coefficient: Using Newton’s binomial formula we have: Theorem. Then: Derivative of a complex function.

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In this lesson we will look at the technique of constructing a sketch of a graph of a function and provide explanatory examples.

Topic: Repetition

Lesson: Sketching the graph of a function (using the example of a fractional-quadratic function)

1. Methodology for constructing sketches of function graphs

Our goal is to sketch the graph of a fractional quadratic function. For example, let's take a function we are already familiar with:

A fractional function is given, the numerator and denominator of which contain quadratic functions.

The sketching technique is as follows:

1. Select intervals of constant sign and determine the sign of the function on each (Figure 1)

We examined in detail and found out that a function that is continuous in an ODZ can change sign only when the argument passes through the roots and break points of the ODZ.

The given function y is continuous in its ODZ; let us indicate the ODZ:

Let's find the roots:

Let us highlight the intervals of constancy of sign. We have found the roots of the function and the break points of the domain of definition - the roots of the denominator. It is important to note that within each interval the function preserves its sign.

Rice. 1. Intervals of constant sign of a function

To determine the sign of a function on each interval, you can take any point belonging to the interval, substitute it into the function and determine its sign. For example:

On the interval the function has a plus sign

On the interval, the function has a minus sign.

This is the advantage of the interval method: we determine the sign at a single trial point and conclude that the function will have the same sign over the entire selected interval.

However, you can set the signs automatically, without calculating the function values, to do this, determine the sign at the extreme interval, and then alternate the signs.

1. Let's build a graph in the vicinity of each root. Recall that the roots of this function and :

Rice. 2. Graph in the vicinity of the roots

Since at a point the sign of the function changes from plus to minus, the curve is first above the axis, then passes through zero and then is located under the x axis. It's the opposite at point.

2. Let's construct a graph in the vicinity of each ODZ discontinuity. Recall that the roots of the denominator of this function and :

Rice. 3. Graph of the function in the vicinity of the discontinuity points of the ODZ

When or the denominator of a fraction is practically equal to zero, it means that when the value of the argument tends to these numbers, the value of the fraction tends to infinity. In this case, when the argument approaches the triple on the left, the function is positive and tends to plus infinity, on the right the function is negative and goes beyond minus infinity. Around four, on the contrary, on the left the function tends to minus infinity, and on the right it leaves plus infinity.

According to the constructed sketch, we can guess the nature of the behavior of the function in some intervals.

Rice. 4. Sketch of the function graph

Let's consider the following important task - to construct a sketch of the graph of a function in the vicinity of points at infinity, that is, when the argument tends to plus or minus infinity. In this case, constant terms can be neglected. We have:

Sometimes you can find this recording of this fact:

Rice. 5. Sketch of the graph of a function in the vicinity of points at infinity

We have obtained an approximate behavior of the function over its entire domain of definition; then we need to refine the construction using the derivative.

2. Solution of example No. 1

Example 1 - sketch a graph of a function:

We have three points through which the function can change sign when the argument passes.

We determine the signs of the function on each interval. We have a plus on the extreme right interval, then the signs alternate, since all roots have the first degree.

We construct a sketch of the graph in the vicinity of the roots and break points of the ODZ. We have: since at a point the sign of the function changes from plus to minus, the curve is first above the axis, then passes through zero and then is located under the x axis. When or the denominator of a fraction is practically equal to zero, it means that when the value of the argument tends to these numbers, the value of the fraction tends to infinity. In this case, when the argument approaches minus two on the left, the function is negative and tends to minus infinity, on the right the function is positive and leaves plus infinity. About two is the same.

Let's find the derivative of the function:

Obviously, the derivative is always less than zero, therefore, the function decreases in all sections. So, in the section from minus infinity to minus two, the function decreases from zero to minus infinity; in the section from minus two to zero, the function decreases from plus infinity to zero; in the section from zero to two, the function decreases from zero to minus infinity; in the section from two to plus infinity, the function decreases from plus infinity to zero.

Let's illustrate:

Rice. 6. Sketch of the graph of a function for example 1

3. Solution to example No. 2

Example 2 - sketch a graph of a function:

We build a sketch of the graph of a function without using a derivative.

First, let's examine the given function:

We have a single point through which the function can change sign when the argument passes.

Note that the given function is odd.

We determine the signs of the function on each interval. We have a plus on the extreme right interval, then the sign changes, since the root has the first degree.

We construct a sketch of the graph in the vicinity of the root. We have: since at a point the sign of the function changes from minus to plus, the curve is first under the axis, then passes through zero and then is located above the x-axis.

Now we build a sketch of the graph of the function in the vicinity of points at infinity, that is, when the argument tends to plus or minus infinity. In this case, constant terms can be neglected. We have:

After performing the above steps, we already imagine the graph of the function, but we need to clarify it using the derivative.

Plotting function graphs. . . . . . . . . . . .
1. Plan for studying the function when constructing a graph. .
2. Basic concepts and stages of function research. . . .
1. Domain of the function D f and set
values ​​of the function E f . Special properties
functions. . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Study of asymptotes. . . . . . . . . . . . . . . . .
2.1. Vertical asymptotes. . . . . . . . . . . . . . .
2.2. Oblique (horizontal) asymptotes. . . . . . .
2.3. Methods for studying non-vertical asymptotes. .
2.4. Relative position of the function graph
and its asymptotes. . . . . . . . . . . . . . . . . . . . . . .
3. Sketching a graph of the function. . . . . . . . . .
4. Sections of increasing and decreasing functions
Minimum and maximum points. . . . . . . . . . . . . . .
5. Convex function up and down
Inflection points. . . . . . . . . . . . . . . . . . . . . . .
3. Differentiation of a function, analytical
whose expression contains a module. . . . . . . . . . . . .
4. Basic requirements for research results
and plotting. . . . . . . . . . . . . . . . . . . . .
5. Examples of function research and construction
function graphs. . . . . . . . . . . . . . . . . . . . . . .
Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Drawing curves. . . . . . . . . . . . . . . . . . . .
1.Plan for research and construction of curves. . . . . . . . . .

2. Basic concepts and stages of curve research. . . . .
	Study of functions x x t and y y t. . . . . . .
	Use of research results x x t . .
2.1. Vertical asymptotes of the curve. . . . . . . . . . .
2.2. Sloping (horizontal) asymptotes of a curve. .
	Analysis of results and construction of a sketch
function graphics. . . . . . . . . . . . . . . . . . . . . .
4. Sections of increasing and decreasing curve
Minimum and maximum points of functions
x x y and y y x , cusp points of the curve. . . . . . .
	Convex function up and down. Inflection points. .
3. Construction of parametrically specified curves. . . . . .
Example 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems for independent solution. . . . . .
Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Graphing functions

1. Plan for studying a function when constructing a graph

1. Find the domain of definition of the function. It is often useful to consider multiple values ​​of a function. Explore special properties of a function: even, odd; periodicity, symmetry properties.

2. Explore the asymptotes of the graph of a function: vertical, oblique. Analyze the relative position of the graph of a function and its inclined (horizontal) asymptotes.

3. Draw a sketch of the graph.

4. Find areas of monotonicity of the function: increasing and decreasing. Find the extrema of the function: minimums and maximums.

Find one-sided derivatives at the discontinuity points of the derivative of the function and at the boundary points of the domain of definition of the function (if one-sided derivatives exist).

5. Find the convexity intervals of the function and the inflection points.

2. Basic concepts and stages of function research

1. Function domain Df and many meanings

tion of the function E f . Special Function Properties

Indicate the domain of definition of the function, mark it on the abscissa axis with boundary points and punctured points, and indicate the abscissas of these points. Finding the domain of definition of a function is not necessary.

It is not necessary to find multiple function values. Easily studied properties of a set of values: non-negativity, boundedness from below or above, etc., are used to construct a sketch of a graph, control the results of the study and the correctness of the graph.

x like

The graph of an even function is symmetrical about the ordinate axis Oy. The graph of an odd function is symmetrical about the origin. Even and odd functions are examined on the positive half of the domain of definition.

A periodic function is studied on one period, and

The chart is shown on 2-3 periods.
2. Study of asymptotes
2.1. Vertical asymptotes
Definition 1.					x x0		called		vertical
asymptote of the graph of the function					y f x,			if completed
one of the conditions:				lim f x 1				lim f x .
				x x0 0				x x0 0
2.2. Oblique (horizontal) asymptotes
noah) asymptote of the graph of the function						y f x at x,
lim f x kx b 0 .
						at x
	definition of asymptote
klim			b lim f x kx . Calculating the corresponding
limits, we obtain the asymptote equation y kx b .
A similar statement is true in the case when
If k 0, then the asymptote is called oblique.
k 0 , then the asymptote				y b is called horizontal.
The concepts of inclined and horizontal are introduced similarly.
asymptotes of the graph of the function y f x							at x.

2.3. Methods for studying non-vertical asymptotes Study of asymptotes for x and for

the rule is carried out separately.

1 We will use the symbol to mean the fulfillment of one case, either

In some special cases, it is possible to jointly study the asymptotes at x and at x, for example, for

1) rational functions;

2) even and odd functions, for the graphs of which the study can be carried out on part of the domain of definition.

Method for selecting the main part. To find the asymptote, select the main part of the function at x. Likewise for x.

The main part of a fractionally rational function It’s convenient to find by highlighting the whole part of the fraction:

Example 1. Find the slanted asymptotes of the graph of a function

f x 2 x 3 x 2 . x 1

f x 2 x 5					o 1 at	x , then straight

May y 2 x 5 is the desired asymptote. ◄

The main part of the irrational function when solving practical examples, it is convenient to find using methods of representing a function by the Taylor formula for x.

Example 2. Find the oblique asymptote of the graph of a function

x4 3 x 1

at x.

x 4 o1

for x, then the straight line				y x 4 is the desired asymptote.
				irrational
				f x 3			convenient to find
	ax2 bx c and				ax3 bx2 cx d

use the method of isolating a complete square or a complete cube of the radical expression, respectively.

Example 3. Find the slant asymptotes of the graph of the function f x x 2 6 x 14 for x and x.

In the radical expression, we select a complete square

x 3 2

5 . Since the graph of the function

f x is symmetrical

relative to straight line x 3 and

then f x ~

at x.

x 3 2 5

So it's straight

y x 3 is

asymptote at x, and straight line y 3 x

Asymptote at

x. ◄

To find asymptotes, you can use the method of isolating the main part.

Example 4. Find the asymptotes of the graph of the function f x 4 x 2 x 2 .

f x 2

That's the function

has an asymptote

y 2 x

and asymptote

y 2 x

at x .◄

For transcendental functions both methods are acceptable

following asymptotes when solving practical examples.

Remark 1. When studying asymptotes irrational, transcendental functions, and functions whose analytical expression contains a module, It is advisable to consider two cases: x and x. A joint study of asymptotes at x and at x may lead to errors in the study. When finding the limits or main part of x, it is necessary to change the variable x t.

2.4. The relative position of the graph of a function and its asymptotes

a) If the function y f x has an asymptote at x,

is differentiable and strictly convex downward on the ray x x 0, then the graph

the fic of the function lies above the asymptote (Fig. 1.1).

b) If the function y f x has an asymptote at x,

is differentiable and strictly convex upward on the ray x x 0, then

the graph of the function lies below the asymptote (Fig. 1.2).

c) There may be other cases of behavior of the graph of a function as it tends to an asymptote. For example, it is possible that the graph of a function intersects the asymptote an infinite number of times (Fig. 1.3 and 1.4).

A similar statement is true for x.

Before studying the properties of convexity of a function graph, the relative positions of the function graph and its asymptotes can be determined by the sign o 1 in the method of isolating the main part.

Example 5. Determine the relative position of the graph

function f x 2 x 2 3 x 2 and its asymptotes. x 1

f x 2 x 5						at x, then gra-
						y 2 x 5 . Because
fic functions lies			above the asymptote

0 at x, then the graph of the function lies below the asymptotic

you y 2 x 5 . ◄

Example 6. Determine the relative position of the graph

functions f x

x4 3 x 1

and its asymptotes for x.

x 2 1

From equality

x it follows that the graph of the function lies below the asymptote y x 4 . ◄

Example 7. Determine the relative position of the graph of the function f x x 2 6 x 14 and its asymptotes.

Since f x x 3 (see example 3), then

x 3 2 5 x 3

the graph of the function lies above the asymptote y x 3 at x and at x. ◄

Example 8. Determine the relative position of the graph

f x 3 x 3 6 x 2 2 x 14 and its asymptotes.

as x 3 6 x 2

2 x 14 x 2 3 14 x 6, then using

a x 2 3 14 x 6 ,

b x 2 3 , we get f x x 2

14x6

3 x 2 3 14x 6 2

x 2 3

x 2 3 14x 6

x 2 2

the difference is positive at x

and negative at x

Therefore, at x, the graph of the function lies below the asymptote y x 2, and at x, above the asymptote y x 2.◄

The method for calculating limits for studying asymptotes does not allow one to estimate the relative position of the graph of a function and its asymptotes.

3. Sketching a graph of a function To construct a sketch of a graph, vertical and

slanted asymptotes, points of intersection of the graph of a function with the axes. Taking into account the relative position of the graph of the function and asymptotes, a sketch of the graph is constructed. If the graph of a function lies above (below) the asymptote at x, then, assuming that

there exists a point x 0 such that among the points x x 0 there are no points of inflection,

we find that the function is convex downward (upward), that is, to an asymptote. Similarly, one can predict the direction of convexity to the asymptote for vertical asymptotes and for the asymptote at x. However, as the above example shows

function y x sin 2 x , such assumptions may not be x

4. Areas of increasing and decreasing function. Minimum and maximum points

Definition 3.	The function f x is called	increasing
(decreasing) on ​​the interval a, b, if for any		x1 , x2 a, b ,
such that x 1 x 2	there is inequality	f x1 f x2
(f x1 f x2 ).

Function f x differentiable on the interval a, b

melts (decreases) on the interval a, b, if and only if

function f x .

A necessary condition for an extremum. If

Point ex-

tremum of the function f x , then at this point either

f x 0 0 , or

derivative does not exist.

Sufficient conditions for an extremum.

f x differential

1. Let there exist 0 such that the function

is radiable in a punctured -neighborhood of the point x 0

and continuous

at point x 0 . Then,

a) if its derivative changes sign minus to plus when re-

progress through the point		x 0 ,
			x x 0 , x 0 , then x 0 is the maximum point
	x 0 for any
functions f x ;
	b) if its derivative changes sign plus to minus when re-
progress through the point		x 0 ,
			those. f x 0 for any x x 0 , x 0 ,
			x x 0 , x 0 , then x 0 is the minimum point
	x 0 for any

functions f x .

Model examples include y x (Fig. 2.1) and